Research on slicing chip forming and wear analysis of arc-tooth slice tool with equal-rake angle

: As a new technology of gear processing, the gear slicing technology has distinct characteristics and broad application prospects. However, the lack in wear and life of the slice tool hinders the further widespread application of the gear slicing technology. Considering the chip forming mechanism is an important basis affecting the wear and service life of the cutter, this paper takes the arc-tooth slice tool with equal-rake angle (ATST-ERA) as the carrier and studies the chip forming and tool wear. Aiming at the particularity of the slicing process, the cutting process is unified and abstracted, and the cutting edge model, the cutting edge race model and the workpiece tooth surface model of the ATST-ERA are established, and then the undeformed chip geometry model is obtained. The correctness of the parametric model of the undeformed chip was verified by comparative analysis. Based on this model, the load of the cutting edge is calculated using the Kienzle empirical formula of the cutting force. By analyzing the relationship with the wear of the rake face, the positive correlation between the cutting edge load and the rake face wear of the ATST-ERA is found. The chip forming modeling method proposed in this paper provides a theoretical basis for the control of the slicing chip forming and the improvement of tool wear.

As a totally new gear machining technology, the gear slicing technology is not realized until the 21st century. The gear slicing technology can not only complete the machining of special structural gears such as thin-walled and non-penetrating internal helical teeth, but also has the characteristics of high efficiency, high precision and environmentally friendly. Therefore, many scholars have studied the gear slicing technology [1][2][3][4][5][6][7][8][9][10][11][12] . Though scholars have achieved breakthrough, the gear slicing technology has not been widely used. The reason is mainly because the slice tool is easy to wear and has a short service life. According to the metal cutting principle, the chip forming law is an important factor affecting tool wear. Because of this, in order to improve the wear performance of the slice tool and improve its service life, it is necessary to conduct an in-depth study on the chip forming mechanism during the slicing process.
Schulze et al. analyzed the chip forming mechanism and its effect on machining reliability by modeling the motion process of the gear skiving [13] . Mazak et al. used the simulation method to study the machining process of the intermittent cutting of the bevel gear. It was found that the peak point of the chip thickness could not be explained from the physical and kinematics, by analyzing the chip shape. Therefore, the method of calculating the chip thickness was proposed to describe the thickness of each part of the chip [14] . Tapoglou established a chip simulation model for power skiving in the CAD simulation environment. Based on this model, the geometry and size of nondeformed chips can be obtained [15] . Pierce et al. used a solid modeling software to create a virtual model of the power skiving process, extracting the chip geometry [16].
The above research is mainly based on simulation techniques, and the relevant research results have not been experimentally verified. In order to carry out in-depth research on the chip forming mechanism during slicing process, we can learn from the related research of chip forming of other machining process, such as turning, milling and hobbing.
Uysa, Su and Zang used the cutting process of stainless steel, graphite cast iron and titanium alloy as the carrier, and studied the chip forming mechanism and threedimensional shape of the chip by means of finite element simulation and cutting experiments. Then they analyzed the effects of cutting speed, feed rate and cutting temperature on the chip shape, and calculated the chip thickness, chip curl radius, shear zone thickness, etc. according to the chip model [17][18][19] . Ahmed et al. considered the influence of tool wear and established the chip model of AISI 304 turning in high pressure cooling environment. It was verified that the model prediction value of chip curl radius was basically consistent with the measured value [20] . Rakesh et al. studied the influence of cutting speed on chip shape by using dry-cutting chrome-nickel alloy as a carrier, and verified it by experiments [21] . Molnar et al. used orthogonal cutting as the research object. By comparing non-stationary cutting and stationary cutting experiments, the influence of cutting thickness on cutting force was analyzed, and the theoretical model related to chip breaking was evaluated [22] .  [23] . Nikawa et al. used CAM simulation software to control the chip thickness by modifying the cutting parameters of the ball-end milling cutter in the NC code, and finally improved the machining ability of the machining method [24] .
Harry et al. proposed a theoretical modeling method for the machining process of turning and milling to predict the geometry of the uncut chip, and then calculated the cutting force, and verified the accuracy of the cutting force calculation result [25] .
Berenji et al. considered the turning and milling process under different machining conditions, studied the influence of machining process parameters on the chip forming mechanism. It was verified that good process parameters improved the machining quality [26] . Bergmann et al. considered the limitations of specific machining methods and specific workpiece materials, and studied the chip forming mechanism during the cutting process of circular cutting edges with different microscopic shapes, and verified them by planning and milling experiments [27] Chen et al. used the gear hobbing as the carrier to establish the space forming model of cylindrical gear of multi-edge interrupted cutting. The chip geometry was obtained by numerical method, and the relationship between the load distribution and wear of the cutting edge was further analyzed [28,29] . Likely, the gear hobbing process is the object. Li et al. used the finite element software to establish the simulation model of gear hobbing of high-speed and dry-cutting, and obtained the deformation law of the chip during the gear hobbing process and the wear state of the hob. The model was validated by the actual gear hobbing experiment [30] . Zhang used the solid modeling method to realize the material removal simulation of the gear hobbing process, and obtained the geometric entity of the undeformed chip. But the research result was not verified by experiments [31] .
As can be seen from the above literature, the motion relationship involved in the gear hobbing process is more complicated than in turning and milling process. In order to study the chips in the gear hobbing process, it is necessary to analyze and simplify the geometry of the gear hobbing process, and to study the chips of the gear hobbing process based on the orthogonal or bevel cutting principle. Similarly, this paper draws on the research method of chips in cutting process such as turning, milling and hobbing, analyzes the machining process of the ATST-ERA, and establishes the parametric model of the gear slicing process and the geometry model of the undeformed chip. The geometry model of the undeformed chip is verified by comparison with the chips obtained from the gear slicing experiment. On this basis, the cutting edge load is calculated based on the empirical formula. By comparing the load distribution of the cutting edge with the wear of the rake face, the positive correlation between the load distribution of the cutting edge and the wear of the rake face is obtained, which further verifies the correctness of the theoretical model established in this paper. The gear slicing technology can realize the machining of various gears such as internal teeth, external teeth, straight teeth and helical teeth. Regardless of the type of gear machining, the slicing process is based on the conjugate rotary cutting principle.
The gear slicing process is similar to the gear meshing process. The tool and the workpiece rotate in a certain speed ratio. The cutting edge that meets the conjugate condition of the line and surface continuously removes excess material during the continuous sweeping process, and finally forms the tooth surface of the workpiece.
According to the principle of conjugate rotary cutting, there are three kinds of motions involved in the gear slicing process, including the rotation of the workpiece, the rotation of the slice tool and the feed of the workpiece or slice tool along the axial direction. In addition, a shaft angle is always maintained between the tool axis and the workpiece axis during the movement of the tool and the workpiece. For the machining of different types of gears, the biggest difference lies in the relative position of the tool and the workpiece, and the cutting amount.
Looking at the tool axis as a reference, if the tool rotates counterclockwise with respect to the workpiece axis, it is considered to be "left pendulum"; if the tool rotates clockwise with respect to the workpiece axis It is considered to be "right pendulum".
When the workpiece rotates counterclockwise with respect to the workpiece axis, the angular velocity is positive, and vice versa; when the tool rotates counterclockwise with respect to the tool axis, the angular velocity is positive, and vice versa.
As shown in Figure 1, when slicing a spur and internal gear, if the cutter shaft swings to the left, the cutter rotates counterclockwise around the tool axis, and the workpiece rotates counterclockwise around the workpiece axis. If the cutter shaft swings to the right, the cutter rotates clockwise around the tool axis, and the workpiece rotates clockwise around the workpiece axis. As shown in Figure 2, when cutting a spur and external gear, if the cutter shaft swings to the left, the cutter rotates counterclockwise around the tool axis, and the workpiece rotates clockwise around the workpiece axis. If the cutter shaft swings to the right, the cutter rotates clockwise around the tool axis, and the workpiece rotates counterclockwise about the workpiece axis. Considering the shaft angle between the tool axis and the workpiece axis, the helix angle of the slice tool must be equal to shaft angle to ensure that the tooth direction of the cutter is consistent with the direction of the tooth groove of the workpiece, so that the tooth groove of the spur gear can be machined.
For the cutting of the spur gear, the difference between the machining of the internal gear workpiece and the external gear workpiece is the center distance value and the tool axis direction, as shown in Figure 3. According to the basic principle of the above-mentioned caries processing, the caries processing process of different types of gears is simplified into a unified motion abstract model, as shown in Figure 4. According to the position and motion relationship between the tool and the workpiece, the coordinate system of slicing is established, as shown in Figure 5. In the figure, the coordinate system S1(o1, x1, y1, z1) is the workpiece coordinate system, o1 represents the coordinate origin of the S1 coordinate system, the unit vectors of x1、y1、 z1 are i1、 j1、 k1, respectively; and the coordinate system S2(o2, x2, y2, z2) is the tool coordinate system, o2 is the coordinate origin of the S2 coordinate system, and the unit vectors of x2、y2、z2 are i2、 j2、 k2, respectively; the coordinate system Sp(op, xp, yp, zp) is The auxiliary coordinate system of the coordinate system S1, op represents the coordinate origin of the Sp coordinate system, the unit vectors of xp、yp、zp are ip、 jp、 kp, respectively; the coordinate system S0(o, x, y, z) is the auxiliary coordinate system of S2, o represents the coordinate origin of the S0 coordinate system, and the unit vectors of x、y、z are respectively i、 j、 k. The coordinate system S1 is used to establish the workpiece tooth surface model, and the coordinate system S2 is used to establish the tool model. The spatial positions of the auxiliary coordinate systems Sp and S0 are fixed and do not change with time. The center distance a represents the vertical distance between the tool axis and the workpiece axis, and the axis angle γ represents the angle between the tool axis and the workpiece axis. φ1 represents the angle of the workpiece coordinate system S1 as the workpiece rotates relative to the initial position, and l represents the distance that the workpiece coordinate system S1 moves relative to the auxiliary coordinate Sp along the workpiece axis (the positive direction of the zp axis), and φ2 represents the angle of the tool coordinate system S2 as the tool rotates relative to the initial position.

Figure 5 Coordinate system
According to the gear machining coordinate system shown in Figure 5, the total transformation matrix of the tool coordinate system S2 to the workpiece coordinate system S1 is: (1) In this formula 11 11 20 cos sin 0 0 sin cos 0 0 0 (4) In the above formula, T20 represents a transformation matrix of the tool coordinate system S2 to the auxiliary coordinate system S0, T0p represents a transformation matrix of the auxiliary coordinate system S0 to the auxiliary coordinate system Sp, and Tp1 represents a transformation matrix of the auxiliary coordinate system Sp to the workpiece coordinate system S1.

Model of edge-sweeping surface
According to the conjugate rotary cutting principle, the cutting edge and the theoretical tooth surface of the workpiece satisfy the conjugate relationship. Since many curves on the conjugate surface of the tooth surface of the workpiece satisfy the conjugate relationship with the tooth surface of the workpiece, a curve that satisfies the conjugate relationship with the tooth surface of the workpiece from tooth root to tooth crest is selected as the cutting edge of the slice tool. Firstly, the conjugate surface of the theoretical tooth surface of the workpiece is obtained, and then a plane or other regular curved surface is used to intersect with the conjugate surface, and the intersection line is the cutting edge curve. The method of deriving the cutting edge curve is basically the same as that used by Wang et al. [12] . It will not be repeated here. The cutting edge curve of the slice tool is shown in Figure 6.

图 6 The cutting edge of the ATST-ERA
The final formed tooth surface of the workpiece is made by multiple cuts of the cutting edge of the cutter, and the motion trace of the cutting edge relative to the tooth surface of the workpiece is the edge-sweeping surface. The cutting edge of the slice tool is usually indicated on the tool coordinate system S2. The tooth surface of the workpiece is usually indicated on the workpiece coordinate system S1. To accurately calculate the position relationship between the edge-sweeping surface and the tooth surface of the workpiece, the coordinate transformation is necessary to be done between coordinate system S2 and S1. In order to facilitate the expression later, the cutting edge on the tool coordinate system S2 is selected to be transformed to the workpiece coordinate system S1. On the workpiece coordinate system S1, the tooth surface of the workpiece is fixed. The cutting edge is fixed on the tool coordinate system S2. After the cutting edge on the tool coordinate system S2 is transformed to the workpiece coordinate system S1, the position of the cutting edge changes with time, and a motion trajectory is formed on the workpiece coordinate system S1, which is the edge-sweeping surface.
The spatial curved surface formed by the cutting edge of any one of the arc tooth cutters swept on the workpiece coordinate system S1 is shown in Figure 7. In the figure, where, j r represents the vector of the point on the j-th edge-sweeping surface on the workpiece coordinate system S1 , r r represents the vector of the point on the cutting edge on the tool coordinate system S2, j indicating the distance between the point on the cutting edge of the j-th edge-sweeping surface on the tool coordinate system S2 and the tool axis, T represents the total transformation matrix from the tool coordinate system S2 to the workpiece coordinate system S1. Let The simultaneous equations (1), (5), (6), (7) can obtain the parametric equation of the edge-sweeping surface as cos sin sin sin cos sin cos 11 where, . x2、 y2、 z2 respectively represent components of the vector of the points on the cutting edge in the tool coordinate system S2 on the x2 axis, the y2 axis, and the z2 axis, all of which are functions of r. Analysis of equations (5) and (8) shows that the mathematical model of the edge-sweeping surface can be seen as a parametric equation for the variables r and  . As shown in Figure 8, the edge-sweeping surface of the arc-tooth slice tool is obtained according to the above formula.
The edge-sweeping surfaces (Σ1, Σ2, ⋯, Σn, n≥3) formed on the tooth groove Ei is the edge-sweeping surface family. Assume that the center angle of the tooth Fi on the slice tool is θ at the time t1. The center angle of the tooth Fk on the slice tool is k       after the workpiece is rotated for n weeks. Where The simultaneous formulae (6) and (11) (12) According to the formulas (1), (11) and (12)  . The edge-sweeping surface family of the arc-tooth slice tool is shown in Figure 9. The cutting amount of the caries processing is very small, and it takes many times to complete the machining of the tooth surface. In order to make the cutting amount as reasonable as possible, the cutting layer in the radial direction is divided into multiple parts, and the layer-by-layer cutting of the workpiece in the radial direction is realized by sequentially adjusting the vertical distance (center distance) between the tool axis and the workpiece axis, until the workpiece tooth surface meets the accuracy requirements.
The slicing of the tooth groove Ei is taken as an example. Figure 10 is a schematic diagram of the sweeping of the tooth slot Ei by two adjacent edge-sweeping surfaces.
It is assumed that the center distance between the arc-tooth slice cutter and the workpiece is a1. Assuming that the cutting amount is δ1, the total cutting amount δ needs to be completed to form the tooth surface of the workpiece that meets the accuracy requirements. Adjust the center distance between the slice tool and the workpiece so that the center distance is a2. Repeat the previous process and the cutting amount that can be completed is δ2. Cycling and reciprocating, when the cumulative amount of cutting reaches δ, the arc-tooth slice cutter completes the machining of the tooth surface of the workpiece to form the final gear. The method of deriving the tooth surface equation is basically the same as that used by Wang et al. [12] .
It will not be repeated here. The tooth surface of the arc-tooth gear is shown in Figure   11. It can be seen from the above analysis that in the cut-in state and the cut-out state, the upper and lower end faces of the workpiece tooth grooves serve as geometric boundaries of the undeformed chips, so that the geometry of the undeformed chips is incomplete and complicated. In addition, most of the cutting conditions during the gear slicing process are completely cut. Therefore, the main research on the undeformed chips in the full cutting state is studied here.
The arc gear is machined by the ATST-ERA as the carrier, and the tool, workpiece and machining parameters are shown in Table 1. Taking a tooth groove Ei of the gear workpiece as an example, during the machining process of the tooth, the p-1 tooth cutting groove Ei-1, the p-th tooth cutting tooth groove Ei, ,the p+1th tooth cutting tooth groove Ei+1., and after the time t, the q-th tooth cutting the tooth groove Ei. With the pth tooth contact slot Ei as the initial time, the q-th tooth starts to contact the slot Ei after the workpiece rotates one revolution, and the required time is t.
According to the geometric relationship, the chip length along the tooth width direction is: In the formula, n1 represents the workpiece rotation speed, v1 represents the workpiece feed speed, and f represents the feed rate.  (16) In the formula, t1 indicates the moment when the j+1th blade sweep surface is just in contact with the tooth surface obtained by the previous cutting process, and t2 indicates the moment when the j-th blade sweep surface is just in contact with the tooth face obtained by the previous cutting process. , t3 represents the moment when the j-th edge sweeping surface is in contact with the finally formed tooth surface.
According to the basic theory of differential geometry, the chip volume corresponding to the spatial path of the cutting edge can be obtained: (17) In this formula, According to the above formula, the shape of the undeformed chips can be obtained by means of the three-dimensional modeling software, as shown in Figure14.

Experimental verification
Using the geometry model of undeformed chip, the undeformed chips with different cutting parameters can be obtained. For the different cutting parameters in Table 2, the undeformed chips got by simulation calculation are shown in Figure 15. In order to verify the geometric model of the undeformed chips, the cutting parameters in Table 2 were selected, and a trial cutting experiment was carried out on the cutting machine tool to collect chips. As mentioned above, for the complete cutting state, the shape of the chips obtained by cutting the workpiece material with the cutting tool teeth is basically the same. Therefore, the chips with common shapes are collected in the experiment as a typical example. The chips produced by different cutting parameters are shown in Figure 16.  As can be seen from Figure 15, the contour of the undeformed chip is a special free curve, and its projection on the plane is approximately crescent-shaped. As for the chip thickness, when the cutting parameters are unchanged, the chip thickness in the cutting-in and cutting-out stages is the smallest, and the chip thickness is the largest in the part corresponding to the boundary line of the blade-sweeping surface. In addition, when the cutting parameters change, the chip thickness increases as the depth of cut increases. Figure 16 shows that the chips collected in the experiment are all deformed chips, and the main deformation form of the chips is curling. The chips after curling are also approximately crescent-shaped. As the cutting depth increases, the chip thickness also shows an increasing trend. Through the above qualitative comparative analysis, it can be seen that the shape of the chip obtained by the simulation calculation is basically the same as the shape of the chip collected by the experiment. The chip shape obtained by the experiment is partially bent or broken which is caused by the extrusion or the friction during the chip-forming process. In addition, the thickness changes of the chips got by the simulation calculation and the experiment are basically the same, and both have a positive correlation with the depth of cut.
In order to quantitatively analyze and compare the simulated and experimental For the undeformed chips obtained by the simulation calculation, the undeformed chips are sectioned in Solidworks, and then the maximum chip thickness is measured; for the deformed chips obtained by the experiment, the curled chips are first spread out, and then with the help of the optical microscope, the thickness value of the deformed chip is obtained by measuring the distance between the upper and lower surfaces of the chip, as shown in Figure 18. For three groups of different cutting parameters, the measured simulation and experimental chip shape parameter values (including: contour radius, arc length, chord length and chip thickness) are shown in Table 3.   According to the empirical formula of cutting load proposed by scholar Kienzle [32] , the cutting thickness and the area of the cutting layer directly affect the cutting load during the cutting process. During the cutting process, the force of the cutting edge against the chip and the force of the chip against the cutting edge are equal and opposite, so the calculation of the load of the cutting edge can be converted into the calculation of the chip load. The chip load can be calculated by the following formula.
Where kc represents the cutting force per unit area when the nominal section thickness and the nominal width of the chip are 1 mm, A represents the area of the cutting layer, h represents the cutting thickness, and u is a constant value coefficient, indicating the influence of the cutting thickness h on the cutting load.
According to the chip forming principle of the gear slicing, the chip formed by the edge-sweeping surface sweeping the workpiece tooth groove reflects the shape of the cutting layer of the cutting edge of the slice tool at different times, as shown in Figure   19. According to the formula 16, the chip thickness s(t,h) of the cutting edge of the slice tool can be extracted at any time, as shown in Figure 20. The contour of the cutting edge of the cutting edge of the slice tool is mainly crescent-shaped. According to the shape of the cutting layer, the area of the cutting layer micro-element at any point on the cutting edge determined by the time t can be calculated as The area of the micro-element at a certain point of the cutting edge is shown in Figure 21. According to the calculation formula of the Kienzle cutting load, the micro-area cutting load at any point on the cutting edge can be derived: Then, the cutting load per unit length of the cutting edge is: In the formula, The linear value of the cutting edge load of the different teeth of the arc tooth cutter can more accurately reflect the comprehensive loading condition of the different cutting edges of the arc tooth cutter. The integrated load of the blade is: Select the parameters in Table 1 to calculate the cutting edge load of the ATST-ERA. The calculated coefficients Kc and u in the Kienzle formula are related to the workpiece material and are usually obtained experimentally. The workpiece material selected here is QT450-10, the parameter value in reference [33] , taking Kc=1451N/mm2, u=0.17. During the machining process of the caries, there are differences in the cutting process of the different teeth of the arc tooth cleaver. In order to get closer to the real situation, the teeth of No. 3, No. 6 and No. 15 of the arc knives are randomly selected for calculation. The calculated load spectrum of the three teeth is shown in Figure 22.
It can be seen that the load distribution has a tendency to gradually decrease toward the both sides in the middle, and the position where the load is the largest is at the left-toleft position in the middle. Considering that the cutting load is the main cause of tool wear, the parameters in Table 2 are selected for the cutting experiment. After testing 100 gear workpieces, the optical microscope of KEYENCE VX-1000 is used to photograph the rake faces of arc cutters No. 3, No. 6 and No. 15, and the wear conditions are shown in Figure 23. It can be seen from the figure that the left part of the cutting edge wears the most severely and gradually weakens to the two sides, and the wear on the left part is more serious than the wear on the right part. Although the number of gear workpieces cut by the cutting tool is small and the wear of the rake face of the cutting teeth is not too serious, there is still crater wear near the cutting edge on the No. 15 cutting tooth, as shown in Figure   24. In order to further analyze the relationship between the cutting edge load distribution and the tool wear, the load spectrum of the No. 15 cutter cutting edge is compared with the rake face wear, as shown in Figure 25. It can be seen from the figure that the wear of the cutting edge in the middle left position is the most serious, and the degree of wear to the left side of the cutting edge is gradually lower, which is also consistent with the average load distribution of the cutting edge. Therefore, the wear amount of the arc tooth cutter is positively correlated with the load distribution of the cutting edge.